# Integrability in the complex case

We introduce the theory of integrable functions in $\mathbb {C}$ . From the integrability of a function it will follow that it is differentiable infinite times, where it is holomorphic.

We will not consider generic integrals, but integrals along paths.

Therefore, we have to define $curves$ and $paths$ in $\mathbb {C}$ .

Definition

A curve in $\mathbb {C}$ is a function $\gamma :[a,b]\rightarrow \mathbb {C}$ such that:

$\forall \,t\in [a,b],\;\gamma (t)\in \mathbb {C} .$ Definition

The set $\gamma ^{\star }=\gamma ([a,b])=\{\gamma (t)\in \mathbb {C} \colon t\in [a,b]\}$ is edfined as the support of the curve $\gamma$ . The points $\gamma (a)$ and $\gamma (b)$ are respectivelty the first and the second extremum of the curve. If $\gamma (a)=\gamma (b)$ we say that the curve is closed.

Definition

A curve $\gamma$ in $\mathbb {C}$ is a path if it is piecewise differentiable.

Now, we can give a meaning to the expression integrate along a path. The definition we will give is the same given in the course of Multivariable Calculus for the elements of an oriented path. All the remarks given in the case of a multivariable function follow: among them, we have the invariance of the integral with respect to the chosen oriented path. In the following, we can take $[a,b]\equiv [0,1]$ .

Definition

Let $\gamma :[a,b]\rightarrow \mathbb {C}$ be a path and $f:{\mathcal {D}}\subset \mathbb {C} \rightarrow \mathbb {C}$ a complex-valued function. We define integral of f along the path $\gamma$ the quantity:

$\int _{\gamma }f(z)dz=\int _{a}^{b}f(\gamma (t))\gamma ^{\prime }(t)dt.$ Definition

We define the length of the curve ( or the path ) $\gamma$ as the quantity:

$L(\gamma )\equiv \int _{0}^{1}\mid \gamma ^{\prime }(t)\mid dt.$ We have that:

$\mid \int _{\gamma }f(z)dz\mid \leq ML(\gamma ),\;{\mbox{where}}M\equiv max\{f(z)\colon z\in \gamma ^{\star }\ .$ An important concept, that distinguishes the integral along paths in $\mathbb {C}$ from the one in $\mathbb {R} ^{2}$ is the index. In particular, we have the following:

Definition

Let $\gamma :[a,b]\rightarrow \mathbb {C}$ be a closed path and $\Omega =\mathbb {C} \setminus \gamma ^{\star }$ . $\forall \,z\in \Omega$ we define index of z with respect to $\gamma$ the quantity:

$Ind_{\gamma }(z)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\xi }{\xi -z}}\equiv {\frac {1}{2\pi i}}\oint dt{\frac {\gamma ^{\prime }(t)}{\gamma (t)-z}}.$ In this defintion we go through $\gamma$ in a counter-clockwise way. We set $Ind_{\gamma }(\infty )=0.$ We can see that every paths $\gamma$ in $\mathbb {C}$ satisfy an important property, analogous to the one given by the Jordan Theorem for simple and closed curves in $\mathbb {R} ^{2}$ .In fact, we have that $\gamma ^{\star }$ always divides $\mathbb {C}$ in two connected components, one bounded and the other unbounded. There is an important relation between the function $Ind_{\gamma }(z)$ and the condition of holomorphism, as given by the following

Theorem

$Ind_{\gamma }(z)$ is an holomorphic function in $\mathbb {C}$ . Moreover, we have that $Ind_{\gamma }\colon \Omega \subset \mathbb {C} \rightarrow \mathbb {Z}$ , that is $Ind_{\gamma }$ is an interger valued function and has constant valued on each connencted component in which $\mathbb {C}$ is splitted by $\gamma$ . On the exterior of $\gamma$ the index is always zero.

As for the definition of interior and exterior, we follow the Jordan Theorem for simple and closed curves in $\mathbb {R} ^{2}.$ We can refer to books of Multivariate Calucus about this topic.

We provide a proof of the previous theorem in a simple case:

Proof

Let $\xi \in \mathbb {C}$ and $\gamma$ be a circumference with centre $z_{0}$ and radius $r$ . So, we have that $\gamma (t)=re^{it}+z_{0},\,t\in [0,2\pi ].$ We obtain that:

$Ind_{\gamma }(z)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\xi }{\xi -z}}={\frac {1}{2\pi i}}\int _{0}^{2\pi }{\frac {\gamma ^{\prime }(t)}{\gamma (t)-z}}dt={\frac {1}{2\pi i}}\int _{0}^{2\pi }{\frac {ire^{it}}{re^{it}}}dt=1.$ We remind the defintions of connected, convex & simply connected set.

Definition

We say that $\Omega \subset \mathbb {C}$ is connected if:
$\forall \,z_{1},z_{2}\in \Omega$ we have that $z(t)=(1-t)z_{1}+tz_{2}\subset \Omega .$ In general, for the other twoue definitions, we refer to textbooks of Multivariate Calculus, where one can find rigorous definitions for $\Omega \subset \mathbb {R} ^{n},\,n\geq 2$ that can be easily extended to $\mathbb {C} .$ We state two of the most important results in complex analysis, given the notion:

${\hat {\mathbb {C} }}\equiv \mathbb {C} \cup \{\infty \}.$ Theorem

We consider $\Omega \subset {\hat {\mathbb {C} }}$ and $f$ an holomorphic function in $\Omega .$ If $\gamma$ is a closed path in $\Omega$ such that $Ind_{\gamma }(z)=0,\,\forall \,z\in {\hat {\mathbb {C} }}\setminus \Omega ,$ then:

$\int _{\gamma }f(z)dz=0.$ Equivalently, we have that:

Let $\Omega \subset {\hat {\mathbb {C} }}$ be a simply connected set (and so ${\hat {\mathbb {C} }}\setminus \Omega$ is connected) and $f$ be a holomorphic function in $\Omega .$ If $\gamma$ is a closed path in $\Omega$ , then:

$\int _{\gamma }f(z)dz=0.$ We see that the theorem provides a necessary condition for the integral along the path to be zero, in fact we could have cases in which the curve has a non-zero index and so we could not say anything about the value of the integral a priori.

An important corollary of Cauchy theorem gives conditions for the comparison between integrals of the same holomorphic $f$ along two different closed paths in $\Omega$ .

Theorem

We consider $\Omega \subset {\hat {\mathbb {C} }}$ and $f$ an holomorphic function in $\Omega .$ Let $\gamma _{1}\;\&\;\gamma _{2}$ be two closed paths in $\Omega$ such that $Ind_{\gamma _{1}}(z)=Ind_{\gamma _{2}}(z),\;\forall \,z\in {\hat {\mathbb {C} }}\setminus \Omega$ , then:

$\int _{\gamma _{1}}f(z)dz=\int _{\gamma _{2}}f(z)dz.$ Proof

Let $f$ be holomorphic in $\Omega$ and $\gamma ,\lambda$ be two closed paths.

We link $\gamma$ and $\lambda$ with two line segments $\pm c$ with a distance $\epsilon$ between them.

It is not possible to have a closed path with zero index with respect to every $z\in {\hat {\mathbb {C} }}\setminus \Omega$ . Let such $\Gamma$ given by: $\Gamma =\gamma \,\cup \,c\,\cup \,-\lambda \,\cup \,-c.$ By applying Cauchy's theorem to the function $f$ considered:

$0=\int _{\Gamma }f(z)dz=\int _{\gamma }f(z)dz+\int _{c}f(z)dz+\int _{(-\lambda )}f(z)dz+\int _{(-c)}f(z)dz=$ $=\int _{\gamma }f(z)dz+\int _{c}f(z)dz-\int _{\lambda }f(z)dz-\int _{c}f(z)dz=$ $=\int _{\gamma }f(z)dz-\int _{\lambda }f(z)dz.$ It follows that: $\int _{\gamma }f(z)dz=\int _{\lambda }f(z)dz.$ 